Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians

274 Pages·1998·8.91 MB·English

by  Viktor S. Kulikov

Encyclopaedia of
Mathematical Sciences
Volume 36
Editor-in-Chief: R. V. Gamkrelidze
Springer-Verlag Berlin Heidelberg GmbH
A. N. Parshin
I. R. Shafarevich
(Eds.)
Algebraic Geometry III
Complex Algebraic Varieties
Algebraic Curves and Their J acobians
Springer
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko,
N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko
Title of the Russian original edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 36, Algebraicheskaya Geometriya-3
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991):
14-02, l4C20, 14C2l, l4C30, l4C34, 14D05, 14D07, 14D20, 14D22,
14F35, 32G20, 14H42, 14H45
ISSN 0938-0396
ISBN 978-3-642-08118-7 ISBN 978-3-662-03662-4 (eBook)
DOI 10.1007/978-3-662-03662-4
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© Springer-Verlag Berlin Heidelberg 1998
Originally published by Springer-Verlag Berlin Heidelberg New York in 1998
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List of Editors, Authors and Translators
Editor-in-Chief
R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Gubkina 8, 117966 Moscow; Institute for Scientific Information (VINITI),
ul. Usievicha 20a, 125219 Moscow, Russia; e-mail: [email protected]
Consulting Editors
A. N. Parshin, I. R. Shafarevich, Steklov Mathematical Institute, ul. Gubkina 8,
117966 Moscow, Russia
Authors
Viktor S. Kulikov, Chair of Applied Mathematics II, Moscow State University
of Transport Communications (MilT), ul. Obraztcova 15, 101475 Moscow,
Russia; e-mail: [email protected] and [email protected]
P.F. Kurchanov, Chair of Applied Mathematics II, Moscow State University of
Transport Communications (MilT), ul. Obraztcova 15, 101475 Moscow, Russia
V.V. Shokurov, Department of Mathematics, The Johns Hopkins University,
Baltimore, MA 21218-2689, USA; e-mail: [email protected]
Translator
I. Rivin, Mathematics Institute, University of Warwick, Coventry CV4 7AL,
United Kingdom, e-mail: [email protected]; Department of
Mathematics, California Institute of Technology, Pasadena, CA 91125, USA,
e-mail: [email protected]
Contents
I. Complex Algebraic Varieties:
Periods of Integrals and Hodge Structures
Vik. S. Kulikov, P. F. Kurchanov
1
TI. Algebraic Curves and Their Jacobians
V. V. Shokurov
219
Index
263
I. Complex Algebraic Varieties:
Periods of Integrals and Hodge
Structures
Vik. S. Kulikov, P. F. Kurchanov
Translated from the Russian
by Igor Rivin
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1. Classical Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
§1. Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
§2. Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
§3. A Comparison Between Algebraic Varieties and Analytic Spaces . 19
§4. Complex Manifolds as c= Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 24
§5. Connections on Holomorphic Vector Bundles . . . . . . . . . . . . . . . . . . 28
§6. Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
§7. Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
§8. Line Bundles and Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
§9. The Kodaira Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
§10. Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 2. Periods of Integrals on Algebraic Varieties . . . . . . . . . . . . . . . 66
§1. Classifying Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
§2. Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
§3. The Period Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
§4. Variation of Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
V. S. Kulikov et al., Algebraic Geometry III
© Springer-Verlag Berlin Heidelberg 1998
2 Vik. S. Kulikov, P. F. Kurchanov
§5. Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
§6. Infinitesimal Variation of Hodge Structures . . . . . . . . . . . . . . . . . . . . 97
Chapter 3. Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
§1. Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
§2. The Cubic Threefold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
§3. K3 Surfaces and Elliptic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
§4. Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
§5. Counterexamples to Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 140
Chapter 4. Mixed Hodge Structures .............................. 143
§1. Definition of mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . 143
§2. Mixed Hodge structure on the Cohomology of a Complete
Variety with Normal Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
§3. Cohomology of Smooth Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
§4. The Invariant Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
§5. Hodge Structure on the Cohomology of Smooth Hypersurfaces ... 168
§6. Further Development of the Theory of Mixed Hodge Structures . . 176
Chapter 5. Degenerations of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . 183
§1. Degenerations of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
§2. The Limit Hodge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
§3. The Clemens-Schmid Exact Sequence ........................ 190
§4. An Application of the Clemens-Schmid Exact Sequence to the
Degeneration of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
§5. An Application of the Clemens-Schmid Exact Sequence to
Surface Degenerations. The Relationship Between the Numerical
Invariants of the Fibers Xt and X0. . . . . . . . . . . . . . . . . . . . . . . . . . . 199
§6. The Epimorphicity of the Period Mapping for K3 Surfaces 205
Comments on the bibliography 211
References 213
Index ......................................................... 261
Periods of Integrals and Hodge Structures 3
Introduction
Starting with the end of the seventeenth century, one of the most interesting
directions in mathematics (attracting the attention as J. Bernoulli, Euler,
Jacobi, Legendre, Abel, among others) has been the study of integrals of the
form l
= r -dz,
Aw(T)
TO W
where w is an algebraic function of z. Such integrals are now called abelian.
Let us examine the simplest instance of an abelian integral, one where w
is defined by the polynomial equation
(1)
where the polynomial on the right hand side has no multiple roots. In this case
the function Aw is called an elliptic integral. The value of Aw is determined
up to mv1 + nv2, where v1 and v2 are complex numbers, and m and n are
integers. The set of linear combinations mv1 + nv2 forms a lattice H C C, and
so to each elliptic integral Aw we can associate the torus C/ H.
On the other hand, equation (1) defines a curve in the affine plane C2 =
{(z,w)}. Let us complete C2 to the projective plane lP'2 = lP'2(C) by the
addition of the "line at infinity", and let us also complete the curve defined
by equation (1). The result will be a nonsingular closed curve E C lP'2 (which
can also be viewed as a Riemann surface). Such a curve is called an elliptic
curve.
C/
It is a remarkable fact that the curve E and the torus H are isomorphic
Riemann surfaces. The isomorphism can be given explicitly as follows.
Let p(z) be the Weierstrass function associated to the lattice H C C.
1 L [ 1 1 ]
p = z2 + (z- 2h)2 - (2h)2 •
hEH,h=/-0
It is known that p(z) is a doubly periodic meromorphic function with the
period lattice H. Further, the function p(z) and its derivative p'(z) are related
as follows:
(2)
for certain constants g2 and g3 which depend on the lattice H. Therefore,
the mapping z --+ (p( z), p' ( z)) is a meromorphic function of Cf H onto the
compactification E' C lP'2 of the curve defined by equation (2) in the affine
plane. It turns out that this mapping is an isomorphism, and furthermore, the
projective curves E and E' are isomorphic!
Let us explain this phenomenon in a more invariant fashion. The projection
(z, w) --+ z of the affine curve defined by the equation (1) gives a double
4 Vik. S. Kulikov, P. F. Kurchanov
covering 1r : E -t IP'1, branched over the three roots z1, z2, z3 of the polynomial
z3 + pz + q and the point oo.
The differential w = dz/2w, restricted to E is a holomorphic 1-form (and
there is only one such form on an elliptic curve, up to multiplication by con
coo
stants). Viewed as a manifold, the elliptic curve E is homeomorphic to the
product of two circles S1 x Sl, and hence the first homology group H1 (E, Z)
is isomorphic to Z E9 Z. Let the generators of H1(E,Z) be ')'1 and ')'2. The
w} .
lattice H is the same as the lattice { m f-r w + n J'Y Indeed, the elliptic
1 2
integral Aw is determined up to numbers of the form fz .j 1 , where l is
z3+pz+q
a closed path in C\ {z 1 , z2, Z3}. On the other hand
l i
J z3 +1 pz + q = w,
where ')' is the closed path in E covering l twice.
J .
The integrals w are called periods of the curve E. The lattice H is called
the period lattice.1fhe discussion above indicates that the curve E is uniquely
determined by its period lattice.
This theory can be extended from elliptic curves (curves of genus 1) to
curves of higher genus, and even to higher dimensional varieties.
Let X be a compact Riemann surface of genus g (which is the same as a
nonsingular complex projective curve of genus g). It is well known that all
Riemann surfaces of genus g are topologically the same, being homeomorphic
to the sphere with g handles. They may differ, however, when viewed as
complex analytic manifolds. In his treatise on abelian functions (see de Rham
[1955]), Riemann constructed surfaces (complex curves) of genus g by cutting
and pasting in the complex plane. When doing this he was concerned about
the periods of abelian integrals over various closed paths. Riemann called those
periods (there are 3g- 3) moduli. These are continuous complex parameters
which determine the complex structure on a curve of genus g.
One of the main goals of the present survey is to introduce the reader to
the ideas involved in obtaining these kinds of parametrizations for algebraic
varieties. Let us explain this in greater detail.
On a Riemann surface X of genus g there are exactly g holomorphic 1-
forms linearly independent over C. Denote the space of holomorphic 1-forms
on X by H1•0, and choose a basis w = (w1, ... ,wg) for H1•0. Also choose a
basis/'= (/'1, ... ,')'2g) for the first homology group H1(X,Z) ~ Z2g. Then
the numbers
are called the periods of X. They form the period matrix Jl = ( Jlii). This
matrix obviously depends on the choice of bases for H1•0 and H1(X,Z). It
turns out (see Chapter 3, Section 1), that the periods uniquely determine the
curve X. More precisely, let X and X' be two curves of genus g. Suppose